Surface cubications mod flips Louis Funar

Louis Funar

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Surface cubications mod flips Louis Funar Institut Fourier BP 74, UMR 5582 CNRS, University of Grenoble I, 38402 Saint-Martin-d'Heres cedex, France e-mail: September 20, 2007 Abstract Let ? be a compact surface. We prove that the set of marked surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with Z/2Z ?H1(?, ∂?; Z/2Z). MSC Classification(2000): 05 C 10, 57 R 70, 55 N 22, 57 M 35. Keywords and phrases: cubication, surface, critical point, cobordism, immersion. 1 Introduction and statements Cubical complexes and marked cubications. A cubical complex is a finite dimensional complex C consisting of Euclidean cubes, such that the intersection of two of its cubes is a finite union of cubes from C, once a cube is in C then all its faces belong to C and each point has a neighborhood intersecting only finitely many cubes of C. A cubication of a topological manifold is a cubical complex that is homeomorphic to the manifold. If the manifold is a PL manifold then one requires that the cubication be combinatorial and compatible with the PL structure. Our definition of cubication is slightly more general than the usual one, because we don't require that the intersection of two cubes consists of a single cube but only a finite union of cubes.

• cubical complexes


• immersion


• higher dimensional


• there exist topological


• boundary


• dimensional manifold


• has


• manifold

Sujets

• Immersion
• Boundary
• Manifold

Informations

• Publié par: pefav
• Nombre de lectures: 84
• Langue: English

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Surface cubications mod ips
Louis Funar
Institut Fourier BP 74, UMR 5582 CNRS,
University of Grenoble I,
38402 Saint-Martin-d’Heres cedex, France
e-mail: [e-mail]
September 20, 2007
Abstract
Let be a compact surface. We prove that the set of marked surface cubications modulo ips, up to isotopy, is in one-to-one
correspondence with Z=2Z H ( ;@ ; Z=2Z).1
MSC Classi cation (2000): 05 C 10, 57 R 70, 55 N 22, 57 M 35.
Keywords and phrases: cubication, surface, critical point, cobordism, immersion.
1 Introduction and statements
Cubical complexes and marked cubications. A cubical complex is a nite dimensional complex C consisting of
Euclidean cubes, such that the intersection of two of its cubes is a nite union of cubes from C, once a cube is in
C then all its faces belong to C and each point has a neighborhood intersecting only nitely many cubes of C. A
cubication of a topological manifold is a cubical complex that is homeomorphic to the manifold. If the manifold is
a PL manifold then one requires that the cubication be combinatorial and compatible with the PL structure. Our
de nition of cubication is slightly more general than the usual one, because we don’t require that the intersection of
two cubes consists of a single cube but only a nite union of cubes.
The study of simplicial complexes and manifold triangulations lay at the core of combinatorial topology. Cubical
complexes and cubications might o er an alternative approach since, despite their similarities, they present some new
features.
nAny triangulated manifold admits a cubication, since we can decompose ann-dimensional simplex inton+1 cubes
nof dimension n. For k = 1 to n we adjoin, inductively, the barycenter of each k-simplex in and join it with the
barycenters of its faces. This way we obtain the 1-skeleton of a cubical complex, as shown in the gure below for
n = 3.
Thus, roughly speaking, working with simplicial complexes is equivalent to working with cubical complexes, from
topological viewpoint. We will show however that cubications might encode extra topological information.
1

It is more convenient to work instead with marked cubications, that is cubications C of the manifold M, which are
endowed with a PL-homeomorphism jCj ! M (called the marking) of its subjacent space jCj. Two markings are
isotopic if one obtains one from the other by right composition with a PL-homeomorphism of M isotopic to identity.
This amounts to give the isotopy class of an embedding of the skeleton of C in M. Marked cubications underlying a
given cubication C are acted upon transitively by the mapping class group of M.
Bi-stellar moves. We will consider below PL manifolds, i.e. topological manifolds endowed with triangulations
(called combinatorial) for which the link of each vertex is PL homeomorphic to the boundary of the simplex. Recall
that two simplicial complexes are are PL homeomorphic if they admit combinatorially isomorphic subdivisions. There
exist topological manifolds which have several PL structures and it is still unknown whether all topological manifolds
have triangulations (i.e. whether they are to simplicial complexes), without requiring them to be
combinatorial.
It is not easy to decide whether two given triangulations de ne or not the same PL structure. One di cult y is
that one has to work with arbitrary subdivisions and there are in nitely many distinct combinatorial types of such.
In the early sixties one looked upon a more convenient set of transformations permitting to connect PL equivalent
triangulations of a given manifold. The simplest proposal was the so-called bi-stellar moves which are de ned for n-
0 0dimensional complexes, as follows: we exciseB and replace it byB , whereB andB are complementary balls that are
n+1unions of simplexes in the boundary @ of the standard (n + 1)-simplex. It is obvious that such transformations
do not change the PL homeomorphism type of the complex. Moreover, U.Pachner ([26, 27]) proved in 1990 that
conversely, any two PL triangulations of a PL manifold (i.e. the two triangulations de ne the same PL structure) can
be connected by a sequence of bi-stellar moves. One far reaching application of Pachner’s theorem was the construction
of the Turaev-Viro quantum invariants (see [31]) for 3-manifolds.
Habegger’s problem on cubical decompositions. It is natural to wonder whether a similar result holds for
cubical decompositions, as well. The cubical decompositions that we consider will be PL decompositions that de ne
the same PL structure of the manifold.
Speci cally , N.Habegger asked ([17], problem 5.13) the following:
Problem 1. Suppose that we have two PL cubications of the same PL manifold. Are they related by the following set
0 0of moves: exciseB and replace it byB , whereB andB are complementary balls (union ofn-cubes) in the boundary
of the standard (n + 1)-cube?
These moves have been called cubical or bubble moves in [10, 11], and (cubical) ips in [4]. Notice that the ips did
already appear in the mathematical polytope literature ([5, 37]).
The problem above was addressed in ([10, 11]), where we show that, in general, there are topological obstructions for
two cubications being ip equivalent.
Notice that acting by cubical ips one can create cubications where cubes have several faces in common or pairs of
faces of the same cube are identi ed. Thus we are forced to allow this greater degree of generality in our de nition of
cubical complexes.
Related work on cubications. In the meantime this and related problems have been approached by several
people working in computer science or combinatorics of polytopes (see [4, 7, 9, 20, 29]). Notice also that the 2-
2dimensional case of the sphere S was actually solved earlier by Thurston (see [30]). Observe that there are several
terms in the literature describing the same object. For instance the cubical decompositions of surfaces are also called
quadrangulations ([23, 24, 25]) or quad surface meshes, while 3-dimensional cubical complexes are called hex meshes
in the computer science papers (e.g. [4]). We used the term cubulation in [10, 11].
Remark that there is some related work that has been done by Nakamoto (see [23, 24, 25]) concerning the equivalence
of cubications of the same order by means of two transformations (that preserve the number of vertices): the diagonal
slide, in which one exchanges one diameter of a hexagon for another, and the diagonal rotation, in which the neighbors
of a vertex of degree 2 inside a quadrilateral are switched. In particular, it was proved that any two cubications of a
closed orientable surface can be transformed into each other, up to isotopy, by diagonal slides and diagonal rotations
2if they have the same (and su cien tly large) number of vertices and if their 1-skeleta de ne the same mod 2 homology
classes. Moreover, one can do this while preserving the simplicity of the cubication (i.e. not allowing double edges).
Immersions and cobordisms. Let M be a n-dimensional manifold. Consider the set of immersions f : F ! M
0 0withF a closed (n 1)-manifold. Impose on it the following equivalence relation: (F;f) is cobordant to (F ;f ) if there
0 0exist a cobordism X between F and F , that is, a compact n-manifold X with boundary F tF , and an immersion
0 :X !MI, transverse to the boundary, such that =ff0g and 0 =f f1g.jF jF
Once the manifold M is xed, the set N(M) of cobordism classes of codimension-one immersions in M is an abelian
group with the composition law given by disjoint union.
Cubications vs immersions’ conjecture. Our approach in [10] to the ip equivalence problem aimed at nding
a general solution in terms of some algebraic topological invariants. Speci cally , we stated (and proved half of) the
following conjecture:
nConjecture 1. The set of marked cubical decompositions of the closed manifold M modulo cubical ips is in bijection
nwith the elements of the cobordism group of codimension one immersions into M .
The solution of this conjecture would lead to a quite satisfactory answer to the problem of Habegger.
Notice that, when a cubical move is performed on the cubicationC endowed with a marking, there is a natural marking
induced for the ipp ed cubication. Thus it makes sense to consider the set of marked cubications mod ips.
We proved in [10] the existence of a surjective map between the two sets.
Digression on smooth vs PL category. There might be several possible interpretations for the conjecture above.
We can work, for instance, in the PL category and thus cubications, immersions and cobordisms are supposed PL.
Generally speaking there is little known about the cobordism group of PL immersions and their associated Thom
spaces, in comparison with the large literature on smooth immersions. However, in the speci c case of codimension-
one immersions we are able to compare the relevant bordism groups. If M is a compact n-dimensional manifold then
PLthe bordism groupN (M) of PL codimension-k immersions up to PL cobordisms is given in homotopy theoreticalk
terms by the formula:
PL 1 1N (M) = [M;
S MPL(k)]k
kwhere PL(k) is the semi-simplicial group of PL germs of maps on R , MPL(k) is a suitable Thom space associated
to it (see e.g. [36]),
denotes the loop space and S denotes the reduced suspe